AG_57.04.14_SADEGHI_finalonline:Layout 6


ANNALS OF GEOPHYSICS, 57, 4, 2014, A0430; doi:10.4401/ag-6407

A0430

Determining precipitable water in the atmosphere of  Iran
based on GPS zenith tropospheric delays

Elaheh Sadeghi*, Masoud Mashhadi-Hossainali, Hossein Etemadfard

K.N. Toosi University of  Technology, Department of  Geodesy and Geomatics Engineering, Tehran, Iran

ABSTRACT

Precipitable water (PW) is considered as one of  the most important
weather parameters in meteorology. Moreover, moisture affects the prop-
agation of  the Global Positioning System’s (GPS) signals. Using four dif-
ferent models, the current paper tries to identify the best relationship
between the atmospheric error known as zenith wet delay (ZWD) and
PW. For that matter, based on 54,330 radiosonde profiles from 11 stations,
two different models i.e. linear and quadratic have been derived for Iran.
For analyzing the accuracy of  these models, ZWDs of  three permanent
GPS stations located in the cities of  Tehran, Ahvaz and Tabriz have been
used. Applying the aforementioned models as well as those already devel-
oped for Europe and the U.S., PWs are derived at the position of  these
stations in Iran. Further, in this research, root mean square error (RMSE)
and bias are the measures for selecting the optimal model. Here, the bias
and the RMSE (between GPS and radiosonde derived PWs) for the pro-
posed linear model for Iran is 1.44 mm and 4.42 mm, and for quadratic
model 2.18 mm and 4.74 mm respectively while, the bias and the RMSE
for Bevis’ linear model is 2.63 mm and 4.98 mm and for Emardson and
Derk’s quadratic models are 2.80 mm and 5.08 mm respectively. As such,
it is observed that the bias of  the proposed linear model for Iran is 1.19
mm and 1.36 mm less than the Bevis’ and Emardson and Derk’s models.
In addition, the RMSE of  the proposed linear model is 0.56 and 0.66 mm
less than the RMSE of  the later ones. This emphasizes that the estima-
tion of  the model coefficients must be based on regional meteorological
measurements.

1. Introduction
Water vapor plays an important role in atmos-

pheric processes. This parameters is one of  the most
variable components of  the Earth’s atmosphere. Its
non-uniform distribution, which is due to the atmos-
pheric phenomena above the Earth surface, depends
both on space and time. Due to the limited spatial and
temporal coverage of  observations, estimates of  water
vapor are always shown with some uncertainties
[Emardson et al. 1998, Jarlemark et al. 1998]. 

Being a key element in various atmospheric phe-

nomena such as precipitation, the amount of  water
vapor is estimated by applying integrated water
vapor (IWV) and precipitable water (PW). In other
words, IWV signifies the amount of  water vapor at a
specif ic direction and the PW is the height of  the
equivalent column of  the liquid water represented by

, where tw is the liquid water density
[Bevis et al. 1992].

The PW is used as one of  the inputs for the numer-
ical weather prediction models [Vedel and Huang 2003] as
well as the pattern analysis and forecast of  rain which is
considered as the most important and fundamental hy-
drological cycle [Seco et al. 2012]. In addition, the PW is
also a useful tool for determining the trends in climate
change [Bevis et al. 1992]. The existing restriction in the
water vapor analysis is a major source of  error in short-
term (0-24 hours) forecasts of  precipitation [Bevis et al.
1992]. A linear power structure has been reported for the
time variation of  the PW [Hogg et al. 1981, Jarlemark et
al. 1998]. The spatial structure of  the PW analyzed by
Emardson et al. [1998] demonstrated the significance of
local models in estimating the precipitable water.

The ground-based GPS stations have improved
both spatial and temporal resolutions of  measure-
ments required for estimating the water vapor
[Rocken et al. 1991]. For this purpose, the propaga-
tion delay experienced by GPS signals in their transi-
tion through troposphere is used. In other words, the
delay in this part of  the atmosphere is partially con-
trolled by the existing gaseous water. Askne and
Nordius [1987] highlighted the required linear rela-
tion between the wet part of  tropospheric delay and
the PW. Over a decade later, Emardson et al. [1998]
revised this ratio for the Scandinavian region. Fol-
lowing years, the same technique was also applied for
estimating the PW in near real time [Karabatic et al.
2011]. Due to their reasonable accuracy, the PWs

PW IWV wt=

Article history
Received September 16, 2013; accepted May 19, 2014.
Subject classification:
Precipitable water, Zenith wet delay, Radiosonde profiles, GPS, Modeling.



were computed using GPS in rain pattern analysis
and forecast models [Seco et al. 2012]. 

Taking into account the ZWDs, the current re-
search primarily intends to assess the necessity and
generate a local (conversion) model for the accurate
estimation of  the PW in Iran. Since radiosondes are
expendable instruments, their high cost often restrict
the number of  launches (normally once to twice a
day at 00,00 and 12,00 in UTC) at each station. It must
be remembered that Iran possesses more than 107
permanent GPS stations hence; the possibility of
computing the PW from ZWDs provides a unique
opportunity for improving both spatial and temporal
resolutions required for estimating the PW in this
country. The paper analyzes the validity of  the exist-
ing models for estimating the PW parameter through
ZWDs in Iran.

The next section discusses the conversion models
which have already been published in other parts of
the world. The paper highlights the development
process of  these models as well as the PW computa-
tion through ZWDs. The third section introduces the
most appropriate model to the climatic conditions of
Iran. For this purpose, the PWs estimated through
different conversion models are compared with the
precipitable water directly extracted from the ra-
diosonde data. The most appropriate model is then
selected using the RMSE and the bias of  GPS and ra-
diosonde derived PWs.

2. Estimating precipitable water from zenith wet
delays

The ZWD of  the GPS tropospheric signals is due
existing gaseous waters in the atmosphere. Taking Π as
the required model for converting the ZWD to the PW,
one can write: PW = ZWD/Π. To estimate ZWD, a slant
tropospheric delay (STD) is computed along the prop-
agation path of  the GPS signals. This parameter is the
signal delay between a satellite transmitting the signal
and a ground station. When the STD is mapped from the
line of  sight to the zenith direction, it is called zenith
tropospheric delay (ZTD). The ZTD is comprised of
hydrostatic and wet components. The ZHD is modeled
using temperature and pressure at the observing site.
The ZWD is computed as the difference between the
observed ZTD and the ZHD [Jin and Luo 2009]:

(1)

Various models are available for computing the
ZHD [Hopfield 1969, Saastamoinen 1972, Goad and
Goodman 1974]. For example, according to Saasta-
moinen [1972]:

(2)

where P0, H0 and { are the surface pressure in hPa, or-
thometric height of  the station in meters and the lati-
tude of  the observing site respectively.

When no pressure and temperature sensors are
available at the GPS station, the data can be extrapo-
lated from the nearest meteorological site. As the ver-
tical temperature gradient is used to transfer the
temperature, Equation (3) can be used to transfer the
pressure [Karabatic et al. 2011]:

where Psyn, Tsyn, hsyn, are the pressure, temperature
and height at meteorological station respectively. Pa-
rameter c is the temperature gradient and R is the gas
constant.

Since, the zenith tropospheric delay is normally
estimated through analyzing the GPS measurements,
a time series of  ZWD can be computed at each per-
manent GPS station. Using the ZWD estimates as
well as the conversion model, the PW amount can be
calculated independently from the radiosonde data
profiles.

As the PW depends on the temporal and spatial vari-
ations of  the atmosphere, the conversion model Π is also
not a constant parameter. Moreover, the best estimate for
this parameter is obtained when the climate condition
is taken into account [Emardson and Derks 2000].

To compute PW from ZWD, there are various ex-
isting models which are classified into two groups of
linear and quadratic here. The following subsections in-
troduce these models in further detail.

2.1. Linear model
The ratio between the ZWD and the PW param-

eters [Bevis et al. 1992] is as followed: 

Here, Rv is the specific gas constant of  water
vapor, that is 461.45 JKg−1K−1, and the atmospheric re-
fractivity constants k3 and are approximately
3.7×105 K2mbar−1 and 17 Kmbar−1 respectively. Tm is the
weighted average of  the atmospheric temperature.
The value of  Tm is calculated using Equation (5).
Therefore, computation of  this parameter at each
point requires water vapor pressure and temperature
along a vertical profile:

ZWD ZTD ZHD= -

0.002277

. .cos

ZHD mhPa

H
P

1 0 0026 2 0 00000028

1

0

0

#

#
{

=

- -

-^ h6 @

P P T
T h h

syn
syn

syn syn
R
g

c
=

- - c^ he o

10PW
ZWD R k T kv m

6
3 2= = +

- l^ h6 @

k2l

SADEGHI ET AL.

2

(3)

(4)



3

In the equation above, T is the temperature in
Kelvin, z is the vertical coordinate and ew is the water
vapor pressure. Further, this parameter is obtained
using the following equation [Holton 2004]: 

Where Lv is the latent heat of  vaporization and is
equal to 2.5×106 JKg−1 and Td is the dew point temper-
ature. To calculate the distance between any two con-
secutive recorded levels, after taking logarithm of  the
equation proposed in [Nafisi et al. 2012] can be used:

Here Pi is the pressure on level i in hPa, Rd is the
gas constant for dry air and is equal to 287.05 JKg−1K−1,
local gravity g is a function of  latitude and height of  the
station and is the mean virtual temperature (Tv) be-
tween the two consecutive levels. The virtual temper-
ature is defined as the one that a hypothetical system of
dry air would have, in relation to the actual condition
of  (moist) air at the same density and pressure. This pa-
rameter is expressed as [Wallace and Hobbs 2006]:

where Mv and Md are the molecular mass of  water
vapor and dry air respectively. Their ratio is defined as
follows [Andrews 2010]:

Tm can be estimated as a linear function of  the tem-
perature at 2 meters high above the Earth surface (Ts)
[Bevis et al. 1992]:

Analyzing more than 8000 vertical profiles of  ra-
diosonde stations in order to compute the unknown
coefficients ai, i = 0,1 in the United States, Bevis et al.
proposed the following functional relation between
Tm and Ts:           

2.2. Quadratic model
According to Bevis et al. [1992], the conversion

model Π and the temperature behave similar to the pa-
rameter Tm with respect to time. Based on the same,

Emardson and Derks [2000] later proposed the follow-
ing functional relation between the conversion model
Π and the difference of  surface and mean surface tem-
peratures:

Where TΔ is the surface temperature minus the
mean surface temperature The mean
surface temperature is computed by averaging the tem-
perature obtained from radiosonde profiles. Further,
the ZWD and PW estimates are required for comput-
ing the unknown coefficients ai,i = 0,1,2. Based on that
as well as using 120,000 radiosonde profiles in Europe,
Emardson and Derks [2000] proposed the following
functional relation [Emardson and Derks 2000]:

To compute the zenith wet delay from the ra-
diosonde profiles, one has [Rocken et al. 1995]:

Here, this integral is computed from the surface
(position of  the station) to the upper troposphere where
T and ew are derived from radiosonde profiles. Practi-
cally, since these parameters are not available in con-
tinuous form, the discrete form of  the integral is used:

In Equation (15), n is the number of  levels in a ra-
diosonde profile. 

Rocken et al. [1995], computed the PW using the
following equation: 

Where tw is the density of  liquid water. This inte-
gral is also computed from the surface (position of  sta-
tion) to the upper most tropospheric layer in the zenith
direction. And for a similar aforementioned reason, the
discrete form of  this equation was used [Hagemann et
al. 2003]:

The PW, for each profile, can be calculated using
the temperature and water vapor pressure derived from
the radiosonde profiles. Then, the PW and ZWD for

T

T
e

dz
T
e

dz
m w

w

2

=
#
#

. .expe T R
L

T6 11 273 15
1 1

w d
v

v

d
= -^ h ; E` j

lnz z g
R

T P
P

i i
d

v
i

i
1

1
- =+

+

c cm m

Tv

T
P M

M
TP

e1
v

d

v
w

=
- -c m

.M
M

R
R

0 62197
d

v

v

d
= =

T a a Tm s0 1= +

. .T T70 2 0 72m s= +

a a T a T0 1 2
2= + +O O

.T T T T= -O^ h

6.458 1.7 10 2.2T T102 25# #= - -O O
- -

10ZWD k T k T e z
Z

Z

w
6

2 3
2

ground

top
d= +- l^ h#

10 2ZWD T T
k

T T
k e e

z z
2

2

i

n

i i

i i

w
i

w
i

i i

6

1

1

1

2

1
2

3
1

1

=
+

+

+
+

+
-

-

=

-

+

+

+

+

l

^

^

h

h

=
=G Ge o

/

PW R T
e

dz1
w v

w
t

= #

PW R T T
e e

z z1
w v i i

w
i

w
i

i

n

i i
1

1

1

1

1t
= +

+
-

+

+

=

-

+< <F F/

PW DETERMINATION FOR IRAN

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)



every profile can be used to calculate the conversion
model Π. Thereafter, unknown coefficients as in Equa-
tion (12) are computed using the corresponding surface
and the mean surface temperatures derived from the
radiosonde profiles.

3. Results and discussion
The proposed linear and quadratic models briefly

introduced in Sections 2.1 and 2.2 are computed first
time for Iran. For this purpose, the current study could
use profiles of  11 radiosonde stations. The acquired
data covers the time interval of  1996 to 2012. These
profiles are obtained from radiosonde lunches which
were made once or twice a day at 00,00 and/or 12,00
in UTC. The corresponding profiles are available for
download in the text format through http://www.esrl.
noaa.gov/raobs/.

Existing gaps in the radiosonde data and the small
number of  lunching stations compared to the much
larger density of  the GPS as well as the continuous data
provided by these equipments emphasize on the ne-
cessity of  developing a suitable conversion model for
Iran. Figure 1 illustrates the position of  the radiosonde
and the GPS stations in this country. 

3.1. Derived models
In the current research, following Section 2, the co-

efficients of  Equations (10) and (12) were locally com-
puted using the data introduced in the previous section.
This section briefly highlights the obtained results. 

3.1.1. Empirical linear model
P in the Equation (4) is only a function of  the pa-

rameter Tm. In order to compute this parameter in the
proposed area of  research, a polynomial model of  de-
gree one, similar to the Equation (10), is used. Here,
54,330 radiosonde profiles have been taken into ac-
count. Further, the unknown coefficients of  the pro-
posed linear model are computed from the inputs given
by the above radiosonde profiles. In other words, the
inputs are the parameters Tm and Ts which are com-
puted and derived respectively from the radiosonde
data. Here, Tm is computed using Equation (5). The ob-
tained polynomial model is given in Equation (18). The
estimated standard deviations for the interception and
slope parameters of  this model are ±0.1 degree of
Kelvin and ±0.003 respectively. 

3.1.2. Empirical quadratic model
In order to find a conversion model of  the second

type (see Equation 12), the ZWD and PW parameters

are computed using the equations given in Section 2.2.
For this purpose, the radiosonde data introduced above
are used. The same number of  radiosonde profiles is
also used for estimating the aforementioned polyno-
mial model (see Equation 18). The estimated model co-
efficients are given in Equation (19) where the root
mean square error stands at 0.09. Moreover, the esti-
mated standard deviations for the parameters of  this
model are ±0.001, ±0.0001 and ±0.00001 for zero, first
and second order coefficients, respectively.

3.2. Computing precipitable water from zenith wet delays
In the current study, for analyzing the accuracy of

the developed models, three GPS stations located in cli-
matologically different areas including Tehran, Ahvaz
and Tabriz were selected. These three cities, respectively,
have semi-dry, wet and semi-wet climatic conditions. 

A comparison of  PWs estimated from ZWDs with
those directly derived from radiosonde profiles helps
analyze the accuracy of  conversion models. In fact,
PWs estimated directly from the radiosonde data are
used as benchmark for comparing the existing and the
proposed models in this research. RMSE and bias are
used to select the optimum result. 

Thereafter, the PW time series obtained from the
GPS zenith wet delays was analyzed using the least
squares harmonic estimation technique. That analysis
provided an insight into the frequency content of  this
time series and therefore could be considered as an inde-
pendent approach for evaluating the proposed models.

To analyze the obtained results, two distinct time

. .T T75 39 0 7103m s= +

6.221 0.01491 6.73 10T T5 2#P = - -O O
-

SADEGHI ET AL.

4

Figure 1. Position of  the radiosonde stations (solid) and the perma-
nent GPS stations in Iran (circles). The position of  the IGS stations
TEHN, AHVZ and TABZ is shown by a solid circles and names here.

(18)

(19)



5

intervals were taken into account with two signified
low and high rainfall, respectively. In fact, the Meteor-
ological Organization of  Iran reported the minimal pre-
cipitation at Tehran, Ahvaz and Tabriz stations in
January, July and September of  2010, 2012 and 2011, re-
spectively. The maximum precipitations were reported
for these stations in May, February and December of
the same years, respectively. As such, the zenith tro-

pospheric delay has been computed for every period
using the GPS measurements of  the above stations. It is
worth mentioning that the distance between the GPS
stations of  this study and the radiosonde stations lo-
cated in Tehran, Ahvaz and Tabriz is 2.3, 36.4 and 11.5
kilometers. The height differences the GPS and the cor-
responding radiosonde stations for Tehran, Ahvaz and
Tabriz are approximately 2.5, 1.0 and 35 meters re-

PW DETERMINATION FOR IRAN

Figure 2 (top to bottom). The RMS and bias of  the proposed as well
as the existing models in the rainy months: i) linear model proposed
in Bevis et al. [1992], ii) linear model proposed in this research, iii)
quadratic model proposed in Emardson and Derks [2000] and iv)
quadratic model proposed in this research.

Figure 3 (top to bottom).The RMS and bias of  the proposed as well
as the existing models in the dry months: i) linear model proposed
in Bevis et al. [1992], ii) linear model proposed in this research, iii)
quadratic model proposed in Emardson and Derks [2000] and iv)
quadratic model proposed in this research.



spectively. Charts 2 and 3 compare the RMS and the
bias of  the models for the GPS stations. 

The analysis of  the obtained results shows that the
models found for Iran perform better than those given
by Equations (4) and (13). Moreover, the RMSE and the
bias for the proposed linear model is less than the quad-
ratic one both for dry and wet months on all of  the
above GPS stations. Figures 4 and 5 illustrate the degree
of  misfit between the proposed linear model and the
PW parameters which are computed directly from the
high resolution radiosonde profiles as well as the time
series of  the estimated precipitable water at the Tehran
station. These figures show the high and the low pre-
cipitations for May 2010 and January 2010, respectively.
Dots are the precipitable water computed from the ra-
diosonde records. 

The frequency content of  the PW time series de-
rived from the GPS zenith wet delays can be considered
as an independent approach for the evaluation of  the
linear model of  this research. The ZWDs of  Tehran,

again a typical station, are used for computing the PW
time series elements. This time series starts at 2010 and
ends at 2012 (see Figure 6)

The frequency content of  this time series can be
analyzed using one of  the existing frequency analysis
techniques. Here, the least squares harmonic estima-
tion has been used [Amiri-Simkooei 2007] because, this
method has been efficiently used for analyzing the fre-
quency content of  various time series i.e. GPS obser-
vations [Amiri-Simkooei 2007], time series of  Total
Electron Content [Amiri-Simkooei and Asgari 2012],
tidal data [Mousavian and Mashhadi-Hossainali 2012]
and coordinate time series for detecting slow slip events
[Mousavian and Mashhadi-Hossainali 2014]. 

The application of  the least squares harmonic es-
timation suggests annual and seasonal frequencies in
the above PW time series. In other words, the annual
and seasonal frequencies are the first that approved by
the hypothesis test of  this method (see Amiri-Simkooei
[2007] for further details). Moreover, since the least
squares harmonic estimation is sensitive to the length
of  the input data [Mousavian and Mashhadi-Hossainali
2012], a detailed analysis of  the frequency content re-
quires a PW time series of  much larger length. Further,
the existence of  annual and seasonal frequencies in the
PW time series indicates that the adopted linear model
does not distort the expected behavior of  the PW pa-
rameter. As a matter of  fact, this can be considered as
independent evidence on the applicability of  the pro-
posed linear model in the field of  meteorology and cli-
matology in Iran.

4. Conclusion
As discussed, the PW is a useful tool for deter-

mining the climate change. It is also used for produc-
ing the rain, as the most important and fundamental

SADEGHI ET AL.

6

Figure 4. Time series of  the estimated pecipitable water at station
Tehran in January 2010.

Figure 5. Time series of  the estimated precipitable water at station
Tehran in May 2010.

Figure 6. Time series of  PW based on the GPS ZWD and Π calcu-
lated using linear model of  Iran during 3 years.



7

hydrological cycle, prediction models. However, re-
striction in spatial and temporal resolution of  the PW
estimates seems to be the major source of  error which
can significantly and reliably be sorted out through the
GPS-derived PW parameters. 

The RMSE and bias of  the proposed models are
slighter than the existing ones. As the obtained results
is valid for both dry and wet climatic conditions in Iran,
the application of  these models seems proper for com-
puting the PW from the ZWDs in this country. In ad-
dition, the empirical linear model investigated in this
research show that the RMSE and bias of  the proposed
model is less than the quadratic one. As such, the ap-
plication of  this model is also efficiently superior for
computing the PW from ZWDs in Iran.

The accuracy analysis of  the PW parameters ob-
tained from the GPS zenith wet delays and the linear
model of  this study, as well as their frequency analysis
approves the reliability of  the GPS-derived precipitable
water. Therefore, the linear model which has been de-
termined here can be used for estimating the PW pa-
rameters together with their direct estimates from
radiosonde profiles. This improved both spatial and
temporal resolutions of  this parameter in the country. 

Acknowledgements. We are grateful to Professor Ghaemi
from the Meteorological Organization of  Iran, for his kind and in-
valuable help during this research. We, particularly, appreciate the
meteorological organization for giving us an access to radiosonde
profiles with dense pressure levels at the stations of  this research.

Data and sharing resources
- Radiosonde data used in this study were taken

from the ESRL radiosonde database, and can be down-
loaded from: http://www.esrl.noaa.gov/raobs

- Zenith Tropospheric Delay of  the IGS was taken
from the Crustal Dynamics Data Information System
(CDDIS) database in the Goddard Space Flight Center
of  NASA, and can be uploaded via: ftp://cddis.gsfc.na
sa.gov/gps/products/troposphere.

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*Corresponding author: Elaheh Sadeghi,
K.N. Toosi University of  Technology, Department of  Geodesy
and Geomatics Engineering, Tehran, Iran; email:
Elaheh_sadeghi@mail.kntu.ac.ir, Sadeghi_elaheh@ymail.com.

© 2014 by the Istituto Nazionale di Geofisica e Vulcanologia. All
rights reserved.

SADEGHI ET AL.

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  >>
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [595.000 842.000]
>> setpagedevice